Completely factor the expression: $$x^8-256$$
Begin by noticing that $256=16^2$.  We can then repeatedly apply a difference of squares factorization: \begin{align*}
x^8-256&=(x^4+16)(x^4-16)\\
&=(x^4+16)(x^2+4)(x^2-4)\\
&=\boxed{(x^4+16)(x^2+4)(x+2)(x-2)}\\
\end{align*}